Oct/Nov 2021

A table is displayed with three column headings: Fraction, Decimal, Percentage. The first row gives: 1/2 = 0.5 = 50. The second row gives: [blank] = 0.25 = [blank]. The third row gives: 1/5 = [blank] = 20.

Fractions, decimals and percentages

Chung places $2460$ in a scheme that pays simple interest at $3.5\%$ each year.

Percentages

A new drink is sampled by 125 people. Each participant assigns a mark out of 5. The bar chart presents the findings. The horizontal axis is titled Score from 0 to 5. The vertical axis is titled Number of people up to 40.

Averages and measures of spread

Find the term that comes next in this sequence.

Sequences

A circular disc has a circumference of 250 cm.

Circles, arcs and sectors

Factorise completely: $18x^2 - 12x$

Algebraic manipulation

Sophie purchases 73 books for her school, and each book is priced at $21.95.

Estimation

Calculate the size of a single interior angle in a regular octagon.

Angles

The table presents the relative frequencies for the outcomes in matches played by a football team. Game result: won, lost, drawn. Relative frequency: won 0.1, lost [blank], drawn [blank]. The number of matches lost is twice the number of matches drawn.

Relative and expected frequencies

Work out $\frac{5}{6}+\frac{2}{5}$ without a calculator. Show every step of your working, and present your answer as a mixed number in simplest form.

Fractions, decimals and percentages

The coordinate axes are labelled $x$ and $y$, and a straight line marked L is drawn on the graph.

Equations of linear graphs

A shaded shape is drawn on a grid made up of 1 cm9 squares.

Area and perimeter

A village has 50 families. $C = \{\text{families who own a car}\}$. $B = \{\text{families who own a bicycle}\}$. 23 families own a car. 10 families own both a car and a bicycle. 6 families own neither a car nor a bicycle.

Sets

The diagram illustrates a sector of a circle with radius 4.8 cm and a sector angle of $45^\circ$. The diagram is not drawn to scale.

Circles, arcs and sectors

A six-sided figure is shown.

Symmetry

A set of shapes is displayed. One T-shaped figure is shaded, and the remaining shapes are labelled A, B, C, D and E.

Transformations

The stem-and-leaf diagram below shows how many items were bought in a supermarket by each of 22 people: 1 | 1 3 6 6 2 | 0 2 2 2 4 8 9 3 | 1 1 5 8 9 9 4 | 2 4 6 7 8 Key: 1 | 1 stands for 11 items.

Averages and measures of spread

Convert 2.7 kilometres to metres.

Units of measure

Hank travels by air from Los Angeles to Shanghai.

Time

$P=2n-3t$

Equations

The scale diagram gives the locations of two towns, P and Q. The scale is 1 cm represents 4 km. A north arrow marked North is shown at each town.

Scale drawings

Write four hundred thousand and four hundred in figures.

Limits of accuracy

State the mathematical name of this polygon.

Geometrical terms

Draw a precise net of this cuboid.

Surface area and volume

Complete the statements below: The modal age is [BLANK]. The median age is [BLANK]. The percentage of women who are older than $51$ years is [BLANK] $\%$.

Averages and measures of spread

Determine the coat’s price in dollars when the exchange rate is $1$ euro $= \$1.15$.

Rates

Find $\begin{pmatrix}3\\-2\end{pmatrix} + \begin{pmatrix}-5\\7\end{pmatrix}$.

Coordinates

Change $2.15$ hours to minutes.

Time

Solve this equation: $7x + 18 = 4$.

Equations

Write down the next term.

Sequences

Find the value for $x$.

Equations

Work out the total number of hours she hires the room for.

Time

What is the value of $\sqrt{345.96}$?

Powers and roots

Work out $\frac{1}{3} - \frac{7}{6} + \frac{1}{5}$. Show all your working, and write your answer as a fraction in its simplest form.

Fractions, decimals and percentages

Find the size of one interior angle in a regular $10$-sided polygon.

Angles

Determine the expected number of left-handed people in a group of $5000$ people.

Relative and expected frequencies

Complete the Venn diagram by filling in the missing values.

Sets

Find the probability that the flower is yellow.

Introduction to probability

Write down the mathematical name for this kind of angle.

Angles

Convert $9\%$ into a decimal.

Fractions, decimals and percentages

State the reciprocal of $20$.

Fractions, decimals and percentages

State the order of rotational symmetry for a rectangle.

Symmetry

Construct a triangle $ABC$.

Geometrical constructions

Determine the temperature gap between midnight and 11 am.

The four operations

Convert $0.3$ metres into centimetres.

Units of measure

Express $\frac{1}{2}$ as a percentage.

Fractions, decimals and percentages

Express 26 g as a percentage of 208 g.

Percentages

The numbers in the list are: 11, 13, 15, 17, 19.

Types of number

Draw a ring around each symbol that makes this statement true: $0.5$ ................. $5\%$.

Fractions, decimals and percentages

The figure shows two parallel lines crossed by a straight line. The angle on the lower line is labelled $132^\circ$, while the angle on the upper line is labelled $x^\circ$. This diagram is not drawn to scale.

Angles

Determine the next term.

Sequences

Sara completes 5 tests. Her average mark is 62. She then completes one further test, and her average mark becomes 68.

Averages and measures of spread

Nina converts 153 euros into dollars at the rate $\$1 = 0.9$ euros.

Rates

A trapezium is drawn. Its upper parallel side measures 7 cm, its height is 12 cm, and its lower parallel side is $y$ cm. Right angles appear at the top right corner and the bottom right corner. The figure is not drawn to scale. This trapezium has an area of 96 $\text{cm}^2$.

Area and perimeter

Marek purchases a computer for $\$420$. He later sells it at a 15% loss.

Percentages

Calculate the radius of a circle whose circumference is 26 cm.

Circles, arcs and sectors

Points $A$ and $B$ are marked on the grid. The $x$-axis is labelled from $-3$ to $3$, and the $y$-axis runs from $-3$ to $4$. Point $B$ appears at the coordinate shown in the diagram, while point $A$ is placed in the lower-left quadrant.

Coordinates

After rewriting each number in the calculation to 1 significant figure, find an estimate for the value of $\frac{4.3 \times 30.7}{6.6 - 1.8}$.

Limits of accuracy

Determine the interior angle of a regular 7-sided polygon.

Geometrical terms

Work out $\frac{11}{12} + \frac{3}{4}$ without a calculator. Show every step of your working, and write your answer as a mixed number in simplest form.

Fractions, decimals and percentages

Simplify $32g^{32} \div 4g^4$.

Algebraic manipulation

Calculate the value of $x$ for this triangle.

Limits of accuracy

How many minutes are there in $4\frac{1}{2}$ hours?

Time

The diagram shows a 3D shape with one circular base, drawn as a dashed ellipse, and sloping sides that narrow to a single apex at the top.

Geometrical terms

Cheng uses a fair 6-sided spinner labelled 1 to 6. The probability scale displayed runs from 0 to 1 in equal intervals.

Introduction to probability

The numbers in the list are: 62, 43, 16, 21, 73, 16, 33, 16, 35.

Averages and measures of spread

The relation $r = 2t + 3u$ is provided.

Equations

At midnight, the temperature stood at $-8^\circ\text{C}$. By noon, it had risen to $6^\circ\text{C}$.

Rates

The chance that it rains tomorrow is $0.47$.

Introduction to probability

$P$ is a prime number with $60 < P < 80$. $P$ is $2$ less than a square number. Determine the value of $P$.

Types of number

A regular polygon has an interior angle of $174^{\circ}$. Determine the number of sides in this polygon.

Angles

Line $L$ is given by $y = 4 - 5x$. Find the equation of a line perpendicular to $L$ that goes through the point $(0, 6)$.

Perpendicular lines

Chai puts some money into an investment. At the end of the first year, the investment's value falls by 35%. By the end of the second year, its value rises by 40% of the amount it had at the end of the first year. Calculate the overall percentage change in the value of the investment.

Percentages

Solve for $x$: $4 - 3x \ge \frac{6 - x}{5}$.

Inequalities

$y$ varies inversely with the square root of $(x - 2)$. For $x = 4.25$, $y = 12$. Determine $x$ when $y = 3$.

Ratio and proportion

The diagram shows three mathematically similar shapes. Their heights are in the ratio small : medium : large = $1 : 5 : 8$. The diagram is not drawn to scale.

Similarity

Find the $n$th term for the sequence $8, 15, 34, 71, 132, \dots$

Sequences

With $y = \frac{3x - 2}{1 - x}$, express $x$ as the subject of the formula.

Algebraic manipulation

The diagram depicts a section of land in the form of triangle $ABC$. $AB = 800\,\text{m}$, $AC = 2300\,\text{m}$ and angle $BAC = 30^{\circ}$. The diagram is not drawn to scale. Houses are to be constructed on this land. Each house needs $400\,\text{m}^2$ of land.

Area and perimeter

Express as one fraction in the simplest form: $\frac{2}{x+3} - \frac{x+2}{7}$.

Algebraic fractions

Hank travels by plane from Los Angeles to Shanghai.

Time

Solve $3(2 + \cos x) = 5$ for values of $x$ on $0^{\circ} \le x \le 360^{\circ}$.

Trigonometric functions

The figure depicts a pyramid $ABCDE$. It has a square base $ABCD$ lying horizontally, with each side measuring 5 cm. The point $E$ is directly above the centre $O$ of the base. The pyramid’s perpendicular height $OE$ is 9 cm. The diagram is not drawn to scale.

Pythagoras' theorem and trigonometry in 3D

Simplify the expression $\dfrac{x^{2/3}}{x^{8/3}}$.

Indices I

Calculate the value of $\dfrac{4.87 - 2.7}{-0.2 + \sqrt[3]{0.729}}$.

Powers and roots

A stem-and-leaf diagram summarises how many items each of 22 supermarket customers bought. It shows: Stem 1: leaves 1, 3, 6, 6 Stem 2: leaves 0, 2, 2, 2, 4, 8, 9 Stem 3: leaves 1, 1, 5, 8, 9, 9 Stem 4: leaves 2, 4, 6, 7, 8 Key: $1 | 1$ means 11 items.

Averages and measures of spread

The table gives the relative frequencies for the games won by a football team. The number of games lost is double the number of games drawn. Complete the table.

Relative and expected frequencies

The scale drawing indicates the locations of two towns, $P$ and $Q$. On the drawing, $1$ cm stands for $4$ km. North arrows are drawn at towns $P$ and $Q$. This diagram is not drawn to scale.

Scale drawings

Do not use a calculator. Find $1\frac{5}{6} + \frac{2}{5}$. Show all your working, and write your answer as a mixed number in its simplest form.

Fractions, decimals and percentages

Solve the pair of simultaneous equations. Show every step of your working. $4x - 2y = -13$ and $-3x + 4y = 11$.

Equations

A right-angled triangle is shown. The sloping side measures 12 cm, the base is 7 cm, and the angle at the base is marked $x^{\circ}$. The diagram is not drawn to scale.

Right-angled triangles

Find the difference between the temperature at midnight and the temperature at 11 am.

The four operations

A regular polygon has an interior angle of $156^{\circ}$.

Angles

A car begins from rest and accelerates uniformly at $0.7\,\text{m s}^{-2}$ for 20 seconds, so that it then attains a steady speed of $14\,\text{m s}^{-1}$. After that, it continues at $14\,\text{m s}^{-1}$ over a distance of $210\,\text{m}$. Next, the car slows down at a constant rate of $1.4\,\text{m s}^{-2}$ until it stops. A speed-time grid is shown.

Graphs in practical situations

The table lists the first five terms in sequences $A$, $B$ and $C$. Sequence $A$: $8, 3, -2, -7, -12$ Sequence $B$: $2, \frac{3}{2}, \frac{4}{3}, \frac{5}{4}, \frac{6}{5}$ Sequence $C$: $\frac{1}{2}, 1, 2, 4, 8$.

Sequences

Write $243 \times 27^{2n}$ in the form of one power of 3, expressed using $n$.

Indices I

On the circle, $P$, $Q$ and $R$ lie on the circumference. $ST$ is tangent to the circle at $R$, and a second tangent from $S$ meets the circle at $V$. The diagram marks angles of $55^{\circ}$ at $P$, $65^{\circ}$ at $Q$, $60^{\circ}$ at $R$, together with $x^{\circ}$ between $QR$ and the tangent.

Circle theorems II

Find the equation of the line joining $A$ and $B$. Write your answer in the form $y = mx + c$.

Equations of linear graphs

Sachin chooses one number at random from the first three multiples of 3, then chooses one number at random from the first three prime numbers. He adds these two numbers to obtain a score. A table is displayed with multiples of 3 as $3$, $6$, $9$ and prime numbers as $2$, $3$, $5$, with some of the scores already filled in.

Conditional probability

Solve for $x$ in $(5x - 3)(2x + 7) = 0$.

Equations

Solve the simultaneous equations below. Show every step of your working. $y = x^2 - 9x + 21$ $y = 2x - 3$.

Equations

There are two Venn diagrams. In the first, the sets $A$ and $B$ are shown, and the highlighted region is $A' \cup B'$. In the second, the sets $C$, $D$ and $E$ are shown, and the highlighted region is $(C \cup D) \cap E'$.

Sets

The stem-and-leaf diagram gives the ages, in years, of 15 women. Stem 3: leaves 1 5 8 9 Stem 4: leaves 1 1 2 3 5 6 9 Stem 5: leaves 0 2 3 8 Key: $3|1$ means 31 years.

Averages and measures of spread

The functions are defined by $f(x) = 2^{x-3}$, $g(x) = 2x - 1$, and $h(x) = \frac{5}{x - 4}$.

Functions

Expand and simplify the expression $(x - 3)^2(2x + 5)$.

Algebraic manipulation

Solve $7\sin x + 2 = 0$ for values of $x$ in the interval $0^{\circ} \le x \le 360^{\circ}$.

Trigonometric functions

Simplify $\frac{3xy + 36y - 5x - 60}{2x^2 - 288}$.

Algebraic fractions

Express $2.15$ hours in minutes.

Time

Triangles $ABC$ and $ACD$ are isosceles. $\angle DAB=86^{\circ}$ and $\angle ADC=58^{\circ}$. The diagram shows $x^{\circ}$ at $\angle ABC$. The diagram is not drawn to scale.

Angles

Angelique hires a room for a party. The charge for the room is $\$15.50$ for the first hour and then $\$7.25$ for every extra hour. In all, she pays $\$95.25$.

Equations

Without a calculator, calculate $\frac{1}{3} - \frac{7}{6} + \frac{1}{5}$. Show every stage of your working and present your answer as a fraction in simplest form.

Fractions, decimals and percentages

Katy has 5 white flowers, $x$ red flowers and $(2x+1)$ yellow flowers. She chooses a flower at random. The chance that it is white is $\frac{1}{12}$.

Introduction to probability

Calculate the value of $\sqrt[4]{\frac{39}{16}}$.

Powers and roots

The numbers given are $2.1 \times 10^{-1}$, $0.\dot{2}$, $22\%$, $\sqrt{0.2}$, and $\frac{24}{1000}$.

Fractions, decimals and percentages

Express 26 g as a percentage of 208 g.

Percentages

The sequence begins with these four terms: 3 -1 -5 -9

Sequences

$P = Mg^{2} + h^{2}$

Algebraic manipulation

Without using a calculator, work out $\frac{11}{12} + \frac{3}{4}$. Show all your working, and present your answer as a mixed number in its simplest form.

Fractions, decimals and percentages

Calculate $0.04^{2} + 0.03 \times 0.28$. State your answer in standard form.

Standard form

A Venn diagram is displayed with universal set $\xi$. Within set $P$ are b and c. Set $Q$ includes e, f and g. The overlap of $P$ and $Q$ contains d. The region inside $\xi$ but outside both circles contains a.

Sets